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Hi Folks --
Here's something fun to think about. Consider the contrast:
a) In electromagnetic radiation, the E-field of a point source
dies off like 1/r. The power and the energy density scale like
E squared, so they scale like 1/r^2, which is consistent with the
idea that energy is conserved as the wave spreads out.
b) In electrostatics, the E-field of a point source dies off
like 1/r^2. The energy density falls off like 1/r^4, but it isn’t
/transporting/ any energy, so conservation doesn’t have anything
to say about it.
The question arises, can we obtain a consistent view of these two
facts? This is not going to be easy, because starting with the
1/r^2 Coulomb field of a point charge, I don’t see any way to
explain the 1/r radiation field. By way of contrast, if there is
some sort of cancellation, I can arrange something that falls off
/faster/ than 1/r^2 – such as a dipole field that falls off like
1/r^3 – but I cannot cook up anything that falls of slower than
1/r^2. I’ve seen a number of books that claim to explain things
this way, but it never made any sense to me.
So some profound questions remain:
a) Can we take the low-frequency limit of the radiation fieldand recover the Coulomb field?
b) Can we wiggle the Coulomb field and get the radiation field?>c) Or are they related in some other way?
The short answers are: (a) no, (b) no, and (c) yes